
theorem Th3:
  for f1,f2 being complex-valued FinSequence holds len(f1 (#) f2) =
  min(len f1, len f2)
proof
  let f1,f2 be complex-valued FinSequence;
  set g = f1 (#) f2;
  consider n1 being Nat such that
A1: dom f1 = Seg n1 by FINSEQ_1:def 2;
  n1 in NAT by ORDINAL1:def 12;
  then
A2: len f1 = n1 by A1,FINSEQ_1:def 3;
  consider n2 being Nat such that
A3: dom f2 = Seg n2 by FINSEQ_1:def 2;
  n2 in NAT by ORDINAL1:def 12;
  then
A4: len f2 = n2 by A3,FINSEQ_1:def 3;
A5: dom g = dom f1 /\ dom f2 by VALUED_1:def 4;
  now
    per cases;
    case
A6:   n1 <= n2;
      then dom f1 c= dom f2 by A1,A3,FINSEQ_1:5;
      then
A7:   dom g = Seg n1 by A1,A5,XBOOLE_1:28;
      min(n1,n2) = n1 by A6,XXREAL_0:def 9;
      hence thesis by A2,A4,A7,FINSEQ_1:def 3;
    end;
    case
A8:   n2 <= n1;
      then dom f2 c= dom f1 by A1,A3,FINSEQ_1:5;
      then
A9:   dom g = Seg n2 by A3,A5,XBOOLE_1:28;
      min(n1,n2) = n2 by A8,XXREAL_0:def 9;
      hence thesis by A2,A4,A9,FINSEQ_1:def 3;
    end;
  end;
  hence thesis;
end;
